Graph Theory

Hitting every large maximal clique with a stable set ★★

Author(s): King; Rabern

Conjecture   There is a universal constant $ \epsilon>0 $ such that every graph contains a stable set which intersects every maximal clique of size $ (1-\epsilon)(\Delta+1) $.

Conjecture   Every graph contains a stable set which intersects every maximal clique of size $ >\frac{2}{3}(\Delta+1) $.

Keywords: independent set; maximal clique

Posted by Andrew King
updated March 31st, 2011
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Graph Theory » Extremal G.T.

Extremal problem on the number of tree endomorphism ★★

Author(s): Zhicong Lin

Conjecture   An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the $ n $ vertices' trees, the star with $ n $ vertices has the most endomorphisms, while the path with $ n $ vertices has the least endomorphisms.

Keywords:

Posted by shudeshijie
updated November 25th, 2020
3 comments
Group Theory

Which lattices occur as intervals in subgroup lattices of finite groups? ★★★★

Author(s):

Conjecture  

There exists a finite lattice that is not an interval in the subgroup lattice of a finite group.

Keywords: congruence lattice; finite groups

Posted by williamdemeo
updated February 22nd, 2011
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Number Theory

Special Primes ★

Author(s): George BALAN

Conjecture   Let $ p $ be a prime natural number. Find all primes $ q\equiv1\left(\mathrm{mod}\: p\right) $, such that $ 2^{\frac{\left(q-1\right)}{p}}\equiv1\left(\mathrm{mod}\: q\right) $.

Keywords:

Posted by maththebalans
updated February 7th, 2013
2 comments
Topology

Hilbert-Smith conjecture ★★

Author(s): David Hilbert; Paul A. Smith

Conjecture   Let $ G $ be a locally compact topological group. If $ G $ has a continuous faithful group action on an $ n $-manifold, then $ G $ is a Lie group.

Keywords:

Posted by porton
updated February 6th, 2011
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